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total_order [2013/09/08 14:55] nikolaj created |
total_order [2014/03/21 11:11] |
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| - | ===== Total order ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $X$ | | ||
| - | | @#FFBB00: $ \le\ \in\ \mathrm{it} $ | | ||
| - | |||
| - | The relation $\le$ is an order relation if it's in the intersection of all total, all anti-symmetric and all transitive relation. Hence | ||
| - | |||
| - | | @#88DDEE: $ \le\ \in\ \mathrm{Rel}(X) $ | | ||
| - | | $ x,y,z \in X $ | | ||
| - | |||
| - | | @#55EE55: $ x \le y\ \lor\ y \le x $ | | ||
| - | | @#55EE55: $ x\le y\ \land\ y\le x \implies (x=y) $ | | ||
| - | | @#55EE55: $ x \le y\ \land\ y \le z \Leftrightarrow x\le z $ | | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Total_order|Total order]] | ||
| - | ==== Parents ==== | ||
| - | === Subset of === | ||
| - | [[Total relation]], [[Anti-symmetric relation]], [[Transitive relation]] | ||
